Mathematical Modeling of Dengue Temperature Effect on Vectorial
Capacity
Jing Helmersson
2012
Supervisor: Joacim Rocklöv
Abstract
Background As climate change and globalization continues, the vector
(mosquito) borne disease - dengue - has changed its pattern, spreading
from tropical and subtropical region to more temperate areas and become
a threat to Europe. Therefore a better understanding of how the
transmission of dengue is affected by climate is an important research
subject in public health.
Objective This study is to develop a theoretical framework in
mathematical modeling of dengue and to explore the relation of dengue
vectorial capacity with temperature – both average and daily variation.
Methods This thesis has reviewed the basic and some sophisticated
theoretical frameworks in mathematical modeling of infectious disease
with focus on dengue modeling. Temperature effect on dengue
transmission was explored from two different models and literature
search for vector and virus transmission parameters. Relative vectorial
capacity for dengue transmission between humans was estimated for
different mean temperatures and diurnal temperature range variation.
Results & Discussion The study showed that the relative vectorial
capacity peaks around mean temperature of 28-30 0C and reduces at both
low and high mean temperature. Large daily temperature fluctuation
increases the dengue transmission at low mean temperature and
decreases the dengue transmission at high mean temperature. As a result,
daily temperature fluctuation reduces greatly the gap in dengue
transmission between warmer and cooler region. As global warming
continues with increased temperature and temperature variation
especially in temperate countries, this result is important in considering
dengue potential and risk assessment based on climate data. Sensitivity
analysis indicated that the mosquito’s mortality rate was the most
important vector parameter in affecting the value of the relative vectorial
capacity especially at low temperatures.
Conclusion This study showed that a simple model can give powerful
insight into the dengue spreading. The generalizability of the model
depends on the vector parameters used. A systematic review of vector
parameters is in great need in mathematical modeling of dengue. Both
choosing the right mathematical model with proper complexity and the
vector parameters are crucial to make modeling useful in understanding,
predicting and guiding dengue control.
Key words: dengue, mathematical modeling, vectorial capacity, daily
temperature fluctuation, Aedes aegypti, climate change, Europe.
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Acknowledgements
This master thesis could not be possible without the understanding and loving support of my
family, Sven, Erik and Lena Helmersson. Special thanks to Professor Ragnar Andersson from
Department of Public Health from Karlstad University, whose suggestion and encouragement
has made my decision to study Public Health and to get a Master’s degree in Umeå
University.
It is not easy to follow other’s ideas after having a Ph.D. in physics and leading a research
group in Physics for over 10 years in California State University Long Beach. It is especially
hard for me to go back to use mathematics after deciding to quit a professor career in physics
in USA. It is my supervisor Joacim Rocklöv’s persistence and gentle encouragement that have
made me choosing mathematical modeling for my thesis and near future research. It is
Lancet chief editor, Richard Horton’s impressive public lecture given in Umeå University
Dec. 2011 with strong emphasis on modeling and its importance in affecting public health
that gave me the enthusiasm in pursuing mathematical modeling. Dr. Horton said as I can
best recall “…Do modeling instead! We, as scientists, need to give politicians and decision
makers confidence by providing scientifically based evidence so that it can make a
difference in public health!” Lancet is the world’s top research journal on general medicine
and public health. Thus, many thanks go to Dr. Joacim Rocklöv and Dr. Richard Horton.
Special thanks to all of my teachers in Public Health in Umeå University, especially Nawi Ng,
for his rich knowledge in epidemiology, great lectures and lessons, hard working and
productive research, which have made me to be humble again. Thanks for all of my
classmates in 2011/2012 public health master’s program for friendship and for
understanding of our differences. Especially thank Charlotte Reding for her spoken of her
heart and gave me a chance to improve myself. Life is a learning process. I thank all of those
who have taught me valuable knowledge and lessons from public health to life along the path
to finish this thesis.
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Content
Abstract .......................................................................................................................................ii
Acknowledgements.....................................................................................................................iii
Content .......................................................................................................................................iv
List of Tables and Figures...........................................................................................................vi
List of Abbreviations and Mathematical Symbols.....................................................................vii
Introduction.............................................................................................................................1
Objective …………………………………………………………………………………………………………………..4
Methods ……………………………………………………………………………………………………………………5
1.
Basic Concepts in Mathematical Modeling of Infectious Diseases
1.1 Mathematical Modeling and Public Health Policy ……………………………………................. 5
1.2 Development of Mathematical Modeling ………………………………………………….…………... 6
1.3 The Basic Concepts in Mathematical Modeling of Infectious diseases…………................ 6
1.4 A Simple Epidemic SIR Model of Infectious Diseases - Humans only…………................. 7
1.5 A Single Epidemic Outbreak, the Reproduction number…..……………………….…………….10
2. Theoretical Frame Work for Dengue Infection
2.1 Dengue transmission process ………….……………………………….…………………….…………... 13
2.2 A SIS Model - Humans and Vectors, the Vectorial Capacity ……………..…….….………….. 13
2.3 Effects of Temperature in Dengue – Seasonality
2.3.1 Sinusoidal Variation of Transmission Rate………….…………………………….…………. 18
2.3.2 A Modified SIR model – Humans, Vectors and Eggs..……………………………………..19
2.4 Effects of Temperature in Dengue – Daily Fluctuation.…………….………………..…….….... 21
Results .....................................................................................................................................24
1.
Combined Effects of Temperature on Dengue Relative Vectorial Capacity …………..….. 24
1.1 Effect of Mean Temperature on Dengue Relative Vectorial Capacity ……….…….….24
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1.2 Effect of Daily Temperature Variation on Dengue Relative Vectorial
Capacity ……………………………………………………………………………………………………… 27
2.
Sensitivity Analysis of Vector Parameters on Dengue Vectorial Capacity ………………….30
Discussion ..............................................................................................................................34
Limitations of this study and suggestions ……………………………………………………………...36
Conclusions ..........................................................................................................................38
References ............................................................................................................................39
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List of Tables and Figures
Table 1
Vector mortality rate (µM) and survival probability (SV).......................................26
Table 2
Comparison of relative change in vectorial capacity V*........................................31
Figure 1
1a: Average Earth surface temperature measured during the 20th and projected
for 21th century; 1b: Aedes mosquito (WHO, 2003)………………….….…….……….....1
Figure 2
Flow Diagram of the SIR model……………………………………….……………………….....8
Figure 3
Flow diagram of dengue infection including humans and mosquitoes……………14
Figure 4
Simulation of a system with sinusoidal transmission rate ………………………...…19
Figure 5
Flow diagram of dengue infection including humans, mosquitoes and eggs…. 20
Figure 6
Vector infection and transmission probabilities as a function of DTR………….. 22
Figure 7
Vector parameters for Aegypti as a function of mean temperature……………..…26
Figure 8
Relative vectorial capacity as a function of mean temperature……….……………. 27
Figure 9
Vector mortality rate as a function of DTR for two dengue viruses….…………….28
Figure 10 Relative vectorial capacity as a function of DTR…………………………………….……29
Figure 11 The effect of different Extrinsic Incubation Period n on relative vectorial
capacity………….……….……………………………………………………………………………...32
Figure 12 The effect of different mortality rate on relative vectorial capacity…..…………...33
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List of Abbreviations and Mathematical Symbols
DENV
Dengue virus
DTR
Diurnal Temperature Range
EIP
Extrinsic Incubation Period
SIR
Mathematical model: Susceptible-Infectious-Recovered
SIS
Mathematical model: Susceptible-Infectious-Susceptible
WHO
World Health Organization
A
Amplitude of annual temperature variation
FI(t)
Seasonal variation in mosquito production from infected eggs
Fs(t)
Seasonal variation in mosquito production from susceptible eggs
IE
Number of Infective mosquito eggs
I, IH
Number of Infective humans
IM
Number of Infective mosquitoes
LM
Number of Latent mosquitoes
NM
Total female mosquito population
NH
Total human host population
R
Number of Recovered humans
Re
Effective reproduction number
R0
Basic Reproduction number
S
Number of Susceptible humans
Sth
Threshold number of Susceptible humans
Sv
The vector survival probability
S*
Endemic number of Susceptible humans
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T
Mean or ambient temperature
V
Vectorial capacity
V*
Relative vectorial capacity, V/m
a
Average number of mosquito bites per person per day
bH
Probability of viral transmission from the mosquito to human per bite
bM
Probability of viral transmission from the human to mosquito per bite
f
Frequency in a yearly cycle of sinusoidal function
g
Proportion of infected eggs laid by infected female mosquitoes
m
Number of female mosquitoes per person
m*
Critical or threshold number of female mosquito per person for dengue transmission
n
Duration of the extrinsic incubation period
p
Vector’s daily survival probability
pc
Fraction of critical vaccination coverage
pI
Per capita infected mosquito egg hatch rate
ps
Per capita susceptible mosquito egg hatch rate
pT
Probability of infection transmission per contact
r
Per capita human birth rate
rM
Oviposition rate of mosquito eggs
t
Time
α, αH
Dengue induced human mortality rate
βH
Viral transmission rate to humans
βM
Viral transmission rate to vectors
λ, λH
Force of infection to humans
λM
Force of infection to vectors
µ, µH
Human natural mortality rate
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µM
Mosquito natural mortality rate
µE
Mosquito egg natural mortality rate
γ, γH
Per capita human recovery rate from dengue
φ
Phase in a sinusoidal function
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Introduction
Global warming is a fact. During the twentieth century, the average Earth surface
temperature increased by approximately 0.6 ºC, of which 0.4 ºC has occurred since 1975 as
shown in Figure 1 below. During this century, the average Earth surface temperature is
expected to increase about 2-30C which exceeds the safe threshold above preindustrial
average temperature (WHO, 2003). The consequences of climate change to health are both
direct and indirect with some being already experienced and others yet to come. For example,
heat waves, extreme weather with consequences, such as flood and other natural disasters,
were already seen, and the changes in the geographical and temporal transmission patterns
of infectious diseases has just started to be observed and more to be expected to come.
Fig. 1. 1a: Average Earth surface temperature measured
th
th
during the 20 and projected for 21 century (WHO, 2003).
1b: Aedes mosquito (WHO, 2012)
As stated in the summary of the book Climate change and human health - risks and
responses (WHO, 2003): “The first detectable changes in human health may well be
alterations in the geographic range (latitude and altitude) and seasonality of certain
infectious diseases – including vector-borne infections such as malaria and dengue fever,
and food-borne infections (e.g. salmonellosis) which peak in the warmer months.”
Dengue is a mosquito-borne viral infection that is usually found in tropical and sub-tropical
regions around the world. According to WHO (2012), “Dengue causes a severe flu-like
illness, and sometimes a potentially lethal complication called dengue haemorrhagic fever”
About 2.5% of those that are infected by dengue die since dengue has neither treatment nor
vaccination.
Dengue has become a major international public health concern. According to WHO (2012),
the incidence of the dengue has increased drastically in the recent decades. For example,
dengue cases increased from 1.2 million in 2008 to over 2.2 million in 2010 across the
Americas, South-east Asia and Western Pacific – 55% increase in two years! Now almost half
of the world population - over 2.5 billion people is at risk from dengue. Dengue
Haemorrhagic Fever or severe dengue was first recognized in the 1950s and has become a
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leading cause of hospitalization and death among children in most Asian and Latin American
countries. WHO currently estimates that every year there may be 50 - 100 million dengue
infections worldwide and about 500 000 people with severe dengue require hospitalization, a
large proportion of them are children. In addition, global mobility and trade have facilitated
dengue spread to areas that were not previously considered important – e.g. Europe, and
potential climate change is likely to contribute to more spread and more suitable conditions
in those areas. In 2010, dengue cases were reported for the first time in France and Croatia,
and imported cases were detected in three other European countries. In Sweden, there are
about 30-60 cases every year from travelers to overseas (Heddini et al., 2009).
Dengue virus is transmitted to human by the two mosquitoes called, Aedes aegypti and Aedes
albopictus. The dengue mosquitoes are typically proliferating in certain tropical and subtropical climate regimens. However since introduced to Europe from Asian through global
trade and travel, the Aedes albopictus mosquito has learned to adapt to temperate climate
and diapauses (overwinter) during the winter season. As global warming continues and
global traffic and trade steadily increases, it is currently spreading northward in Europe and
is even expected to reach Sweden in the year of 2030s (ECDC Technical Report, 2009).
However, our understanding of the dengue transmission overall, and in particularly in
temperate climate regimes is very limited, especially the effect of temperature and
temperature daily variability.
Therefore a better understanding of how the transmission of dengue is affected by climate
especially temperature and its variation is an important research subject in public health. It is
important to monitor and model the spread of dengue to vulnerable areas where people have
no or little immunity, such as, Europe. Mathematical modeling can help our understanding
and assessment of the present and future risk areas on spread of infectious diseases based on
climate data as shown in the case of the malaria cartography (Gething et al., 2011).
Mathematical modeling uses a set of mathematical equations derived from a theoretical
framework and calculates the threshold condition such as, the vectorial capacity for
transmitting virus and/or incidences of dengue as a function of time for a particular area. In
other words, mathematical modeling can help us not only understand and predict the future
spread of infectious diseases but also evaluate strategies on combating dengue (Burattini et al
2008). Using computer simulation from mathematical modeling one can produce estimates
of disease transmission, e.g. disease incidences under certain assumptions, and threshold for
epidemic outbreaks.
The accuracy of modeling a real situation depends on the assumptions of the theoretical
framework and parameters used to describe the relations between human and mosquito
populations, mosquito and virus interaction in the virus transmission and disease spreading
process. The first step for modeling dengue is to develop a good theoretical framework to
describe the dengue transmission in a given environment. The theoretical framework should
capture all the key variables and make approximations on other less important variables.
Temperature is a key environmental determinant in shaping the landscape of dengue, while it
is often not incorporated explicitly in disease models at the present, especially daily
temperature variation. However, diurnal temperature variation can be higher in temperate
countries compared to tropical countries and is therefore essential to be incorporated in
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order to understand dengue transmission and quantify risks in vulnerable areas such as
Europe.
Today most studies on modeling of infectious diseases are based on theoretical frameworks
that consider only constant or average temperature (Macdonald, 1952; Diekmann et al.,
1990). As shown recently both the daily (Lambrechts et al., 2011) and seasonal (Massad et al.,
2011) temperature fluctuations have important impact on some factors in the transmission of
the dengue virus. However, both studies have either neglected or treated the other vector
parameters as temperature independent in the dengue transmission. Thus, there are no
studies in the dengue modeling taking into account of the temperature effect of all the
important parameters in the chain of events of causing dengue transmission.
Within the DengueTools research program carried out in the department of Public Health at
Umeå University as part of the global project, studies are conducted to better understand the
risks of dengue infection in Europe through both empirical and modeling studies. This
Master thesis is the part of the modeling effort of the program that intends to fulfill part in
one of the three gaps described in DengueTool project (Annelies et al., 2012):
“3. Lack of predictive models for the risk of establishment of dengue in uninfected regions
(in particular Europe), taking into account global travel networks and climate change”
Thus, this thesis intends to review and develop a theoretical framework for dengue
mathematical modeling and to estimate the potentials risk to vulnerable areas with focus on
Europe.
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Objective
The objective of this thesis is to develop a general theoretical framework for mathematical
modeling of dengue transmission potential based on temperature. Through reviewing the
existing dengue mathematical models, this study aims at finding the best way to incorporate
temperature effect on dengue transmission. Through reviewing vector data, it intends to
specify the vectorial capacity as a function of the daily average temperature and daily
temperature variations experienced to an entirely susceptible population.
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Methods
This section reviews the commonly used mathematical modeling frameworks for infectious
diseases first. Then it will zoom in to dengue modeling specifically. Emphasis will be given to
those dengue models that incorporate temperature effect. These are bases for developing the
theoretical framework for this thesis. The results of the modeling are generally obtained
numerically through sophisticated computer programming. For the scope of this Master’s
thesis, mainly analytical solutions are presented.
1. Basic Concepts in Mathematical Modeling of Infectious Diseases
1.1 Mathematical Modeling and Public Health Policy
The goal of public health research is more than just knowledge quest. It aims at making a
difference in public health. The frustration that faced many scientists in public health is that
their research results were not taking into account in the decision making process. That is
why we have making health policy as part of our curriculum in public health master program.
In public health, we may divide the research into two types of studies: empirical and
mathematical modeling. Empirical study consists of 1) designing the study – quantitative or
qualitative, 2) collecting data, 3) analyzing data and 4) reporting results as seminars and
publications. This is the main stream of study in public health. In the last few decades,
modeling starts to enter in public health. Its sophistication in attacking research problems
increases with the capacity of computer’s development. Modeling consists of 1) developing a
theoretical framework – transfer a research problem into a set of mathematic equations, 2)
finding relevant parameters to connect with health reality from empirical data, 3) computing
the solutions of the equations in numerical and/or graphic form, and 4) reporting results as
seminars and publications. One of the important differences between these two study types is
the relevant time frame at focus: empirical study is data based and data is from events that
happened before. Thus the focus of study is what has occurred in the past. One may predict
future events with limitations to the same conditions as in the past events. Whereas,
mathematical modeling focuses directly on the prediction of past or future events based on a
set of assumptions and past events data and projected future conditions. A mathematical
disease model constitutes a set of causal pathways involved in the exposure to disease process
and simulates disease transmission over time and space.
The importance of mathematical modeling in public health policy is strongly stressed by the
Lancet chief editor, Richard Horton, during his public lecture given in Umeå University 2011
(Horton, 2011).
In fact, Mathematical (computer) modeling has been used in evaluating social and economic
policies. It can be used to evaluate health policies as well. As quoted by Aron J. (2007):
“Properly used, computer models can improve the mental models upon which decisions are
actually based and contribute to the solution of the pressing problems we face.”
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Although it is not easy to include all the socioeconomic and demographic factors beside
climate factors, mathematical modeling gives an tool to facilitate consensus and action with
an iterative and incremental approach to making decisions (Aron J., 2007).
1.2 Development of Mathematical Modeling
Using mathematical language to describe the transmission and spread of infectious diseases
is not new (Kretzschmar and Wallinga, 2010). As early as 1766, Daniel Bernoulli used
mathematical life table analysis to describe the effects of smallpox variolation (a precursor of
vaccination) on life expectancy (Dietz and Heesterbeek, 2000). However, only in the
twentieth century, the nonlinear dynamics of infectious disease transmission was better
understood. Hamer (1906) was one of the first to recognize that the decreasing density of
susceptible persons alone could stop the epidemic. The 1902 Nobel prize winner, Sir Ronald
Ross, developed mathematical models to investigate the effectiveness of various intervention
strategies for malaria. In 1927, Kermack and McKendrick derived the celebrated threshold
theorem (Kermack & McKendrick, 1927). They found that a threshold quantity is needed in
order for an infectious disease to spread in a susceptible population. This leads to the concept
of herd immunity. The threshold theorem has been very valuable during the eradication of
smallpox in the 1970s (CDC, 2012).
Mathematical modeling became more widespread toward the end of 20th century. Especially
in public health policy making at strategic and tactical levels, modeling approaches have the
advantage of predicting the future courses of an epidemic and the consequences associated to
different scenarios to identify the most effective preventions strategies (Anderson & May,
1991). Such models have successfully been used to better surveillance and control of the AIDs
pandemic in 1980-90s, the UK Foot & Mouth disease livestock epidemic 2001 (Keeling M.J.,
2005) and the outbreak of the SARS virus 2003 (Wallinga and Teunis, 2004).
1.3 The Basic Concepts in Mathematical Modeling of Infectious diseases
Models help us to understand reality because they simplify it. In a sense, a model is always
“wrong” since it is not reality (Aron J., 2007). However, a model may be a useful
approximation, permitting conceptual experiments that would otherwise be difficult or
impossible. Models need to capture essential behavior of interest and incorporate essential
processes. Making models explicit mathematically clarifies thinking and allows others to
examine them. Thus mathematical models allow precise, rigorous analysis and quantitative
prediction.
Therefore, it is not necessary that the more complex model is always the best. Complicated
and detailed models are usually better in fitting the data than simpler models, but they can
obscure the understanding and the mechanism responsible for the result. Simpler models are
more transparent which provide insight and guide thought better (Aron J., 2007). “The
choice of the optimal level of complexity obeys a trade-off between bias and variance
(Burnham & Anderson, 2002). A model should only be as complex as needed, depending on
the questions of interest. This philosophy is referred to as Occam’s razor or the principle of
parsimony and can be summarized as the simplest explanation is the best” (Choisy et
al., 2007).
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There are two types of mathematical models: deterministic (or transmission) and stochastic
(or statistical). Deterministic models are those in which there is no element of chance or
uncertainty. As such, they can be thought to account for the mean trend of a process and
more suitable for disease propagation in large population. Stochastic models, on the other
hand, account not only for the mean trend but also for the variance structure around it. This
is proper when the population is small and random events cannot be neglected (Choisy et al.,
2007). In this thesis, I will focus only deterministic model assuming that a large population
will be our concern for the dengue infection.
In a deterministic model, the time evolution of an epidemic is described in mathematical
terms. It connects the individual level process of transmission with a population level
description of incidence and prevalence of an infectious disease. Thus, modeling builds on
our understanding of the transmission process of an infection in a population, such as, the
prevalence of infectious individuals, the rate of contact between individuals, infectiousness of
the infected individuals or vectors, etc. Thus, the following factors are important:
1) population demography, e.g. age, sex, population density, birth and death rate;
2) natural history of infection, e.g. latency, infectious period, immunity;
3) transmission of infection, e.g. direct or indirect, contact rate.
The transmission process is generally a dynamic process where the individual’s risk of
infection can change over time. It requires dynamic models, e.g. modeling state variables as a
function of time, and can be used for prediction and analysis of disease spread and
preventative programmes.
Modeling generally consists of four steps. First, a flow diagram represents the natural history
and transmission of infection. Based on this diagram, we can then write a set of mathematical
equations to express the transmission process. The third step is to find proper values for the
parameters used in the equations. Finally, we need to solve the equations algebraically or
numerically with help of computer simulation programs. Thus, mathematical modeling
involves many disciplines, from clinic medicine, biology, mathematics/physics, computer
science, zoology, to environmental and social science. Due to short time limit of this thesis, I
will focus on the first three steps and carry out the last step of calculation only for simple
cases.
1.4 A Simple Epidemic SIR Model of Infectious diseases – Humans Only
Let us consider first a simple case of transmission process of an infectious disease
(Kretzschmar and Wallinga, 2010), such as measles. As mentioned earlier, some of the most
important quantities can be learnt from simple models. Analysis of this model helps us to
develop a deeper understanding of the phenomenon of epidemic spread and disappearance.
Here individuals in the population can be classified into three compartments:
a) Susceptible to the disease (Susceptibles)– S,
b) Currently Infectious (Infectious or Infectives) – I,
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c) Recovered and immune (Recovered or Removals) – R.
Here S, I, R, represent the number of individuals in each compartment. The total host
population is N = S + I + R. The transmission process is represented by the flow diagram
shown in Figure 2. This is called SIR model. Here we have neglected details including vectors
and considered just human population.
Susceptible
Infectious
Recovered &
Immune
S
I
R
birth, r
Recovery, γ
Infection, λ
death, µ
death, µ, α
death, µ
Fig. 2. Flow Diagram of the SIR model
Each arrow represents the flow rate at which individuals enter or leave a compartment per
unit time, that is, the incidence rate. The number of susceptible individuals (S) is increased
by birth (rate r) and decreased by natural (non-diseased) death (rate µ) and by transmission
of infection events of susceptible (rate λ, the force of infection). The number of Infectious
individuals (I) is increased by infection events of susceptible, and decreased by natural death
(rate µ) plus disease-induced death (rate α) and by recovery (rate γ) of infected individuals
into immunity. The number of recovered individuals (R) is increased from recovered
infectious individuals and decreased by natural death (rate µ).
All the parameters r, µ, α, λ and γ are per-capita rates. The population flow rate is the percapita rate (r, µ, α, λ or γ) multiplied by the number of individuals subjected to that percapita rate (N, S, I, or R). For example, in-flow “birth” to compartment of “susceptible” is rN,
the per capita birth rate r times the total population N of the system, assuming all individuals
give birth at the same rate r which is averaged over males and females. The population
recovery rate is γI, the per capita recovery rate γ times the number of infected I. Thus, the
population flow rate as each arrow shown in the flow diagram is summarized below:
Birth = rN.
Infection = λS,
Recoveries = γI.
Deaths of S = µS.
Deaths of I = (µ+α)I.
Deaths of R = µR.
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The next step is to write equations based on the flow diagram to express change of state
variables: dS/dt, dI/dt, dR/dt. Here t is time.
dS/dt = birth – Infection – deaths (of S)
dI/dt = Infection – Recoveries – deaths (of I)
dR/dt = Recoveries – deaths (of R)
Every term is a rate quantity and has the unit of 1/time. Using the parameters and population
flow rates described above, we have the mathematical expression of the transmission process
in form of differential equations:
dS/dt = rN – λS – µS
(1-1)
dI/dt = λS – γI – (µ+α)I
(1-2)
dR/dt = γI – µR
(1-3)
N=S+I+R
(1-4)
Here the only non-constant parameter is λ, the force of infection, which is the per-capita rate
of infection of susceptible and describes the risk that a susceptible individual will get infected
per unit time. λ depends on the rate of contact with other individuals, c, the probability of
transmission when an infectious individual contacts a susceptible, pT, and the infectious
proportion in the population, I/N . λ can be expressed as:
λ = pT c I/N, or
= βI/N.
(1-5)
where β=pT c is the total transmission rate in the population. Put equation (1-5) into equation
(1-1 & 1-2), we have a set of four non-linear differential equations, Eqs. (1-1)-(1-4) for the
transmission process.
Here S, I, R, N are state (or dependent) variables that change with time. They describe the
state of an epidemiological system – population in each compartment in this case. Their
dependence on time varies intrinsically and can be simulated in model using a computer
program based on the equations given (1-1) – (1-5). They are not manipulated directly. On
the other hand, r, µ, α, β (or c & pT) and γ are parameters that do not change with time. Once
their values are specified, they stay constant during the calculation as the computer program
runs. They are chosen either based on estimates from epidemiological data, or based on
assumptions.
Assume that the values of parameters are specified based data and assumption made for a
specific infectious disease and a specific population, we move to the final step of modeling –
solving the equations. Solving the equations will give us the time evolution of state variables.
Most of the models cannot be solved algebraically. Numerical integration using a computer is
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normally the standard methods. We need to specify an initial state and the computer will run
the program to solve the set of equations (1-1) – (1-5) as an iterative procedure over time.
However, under special conditions, some of the analytical solutions can be obtained. Here,
some of these analytical solutions will be listed, especially those related to the threshold
conditions.
1.5 A Single Epidemic Outbreak, the Reproduction Number
We can gain some insight and learn the most important concept from the SIR model on a
simple case – a single epidemic outbreak. During a single epidemic outbreak, the time span is
normally short so that any demographic change can be neglected (Iannelli 2005). That is, we
assume that no birth and death occur during this time. In other words, the birth and death
rate are zero on the scale of average duration of infectivity (1/γ), or µ+α << γ. The total
population (N) is constant. In addition, we assume that the whole population N is susceptible
at the beginning:
S0 = N, where S0 is the initial value of S at time t=0.
Eq. (1-2) can be rewritten while putting in Eq. (1-5) as:
dI/dt = λS – γI
= βSI/N – γI
= (βS/N – γ)I
= γ(Re(t) − 1)I
(1-6)
Where Re(t) is defined as:
Re(t) = βS/(Nγ) = R0 S/N,
(1-7)
and R0 is defined as
R0 = β/γ .
(1-8a)
Here, Re(t) is known as the effective reproduction number and R0 is known as the basic
reproduction number. They are sometimes called reproduction ratio or reproduction
rate (Aron J., 2007). Since both β and γ have the same unit of 1/time, Re and R0 are
dimensionless numbers. Either number or ratio are accurate description for this quantity.
Here this thesis adopted the conventional name – reproduction number, although
reproduction ratio is just as good or may be more clear in term of its meaning as explained
later.
In order for an epidemic to take place, the number of infected persons (I) must increase. This
means mathematically that
dI/dt > 0, or Re(t) > 1.
- 10 -
Eq. (6) also shows that
if Re(t) = R0 S/N < 1, then dI/dt < 0 - the number of infectious population is decreasing.
If Re(t) = R0 S/N = 1, then dI/dt = 0.
(1-9)
dI/dt = 0 or Re(t) = 1 means that the number of infectious (I) reaches its maximum or stays
constant. This is the threshold value. From Eq. (1-9), the threshold number of susceptible
(Iannelli 2005) is needed to sustain the infection is when R0 S/N = 1 or
Sth = N/R0
(1-10a)
At the beginning when a new case was introduced to a totally susceptible population (S0 = N),
the transmission process can be described by Re(t=0) = R0. If R0 > 1, the rate at which
susceptibles become infectives exceeds the rate at which infectives are recovered, or the
number of new infectious increases at first. As time goes on, a part of the population is
infected and become immune. The number of available susceptible individuals (S) decreases.
As S becomes less than the threshold value Sth = N/R0, Re(t) become less than 1. The
epidemic dies out as it runs out of susceptible individuals. If initially the population is only
partially susceptible due to intervention or immunity from past recovery of the same disease,
still Re > 1 is needed to have the epidemic spread and Re < 1 is the criterion for the epidemic
to stop.
Both R0 and Re are threshold quantities that predict the occurrence of an epidemic. The value
of either Re or R0 is greater than or less than one respectively determine the prevalence of
infection to increase or decrease for a totally (R0) or partially (Re) susceptible population
(Cintron-Arias et al., 2009). While the basic reproduction number (R0) is defined by
Macdonald (1952) as the number of secondary infections produced by a single
infective in a completely susceptible population, the effective reproduction number
(Re) describes the number of secondary cases produced per index case in a
population that is only partially susceptible. Since S ≤ N, Re (t) ≤ R0. The effective
reproduction number Re depends on the state variable S/N and is changing with time which
makes it less easy to be used in the prediction of an epidemic without computer simulation.
On the other hand, the basic reproduction number R0 is a constant that depends on
parameters only. From Eq (1-8a), we see that
R0 = β/γ = pT c/γ.
(1-8b)
As shown, the basic reproduction number is determined by three parameters: the average
rate of contact between susceptible and infected individuals (c), the probability of infection
being transmitted during a contact (pT), and the duration of infectiousness (1/γ). R0 is the
central quantity in infectious disease epidemiology. R0 defines the threshold value for an
epidemic to occur in a completely susceptible population. If R0 > 1 or a single case introduced
into a susceptible population generates more than one new case, the number of cases is
increasing and an epidemic will spread. If R0 <1 or a single case introduced into a susceptible
population generates less than one new case, the number of cases decreases and an outbreak
- 11 -
will die down. However, to successfully eliminate a disease from a population, the effective
reproduction number Re <1 needs to be maintained even for a completely susceptible
population initially.
R0 can in principle be determined for every infectious disease based on model and can be
estimated for every infectious disease although it is not easy in the beginning of an epidemic
(Iannelli 2005).
The great contribution of the SIR model is that the value of R0 and Re can be used to analyze
the dynamics of transmission, both when infection has just been introduced (R0) into a
population and when infection has long been endemic (Re). Endemic means that the chain of
transmission from infective host to susceptible host is maintained in a population or
infectious population reaches an equilibrium constant:
dI*/dt = 0, or Re(t)* = R0 S*/N = 1.
Here endemic state variables are labeled with *.
This means that S* = N/R0, or
S*/N = 1/R0
(1-10b)
Thus, at endemic state, a single case will generate only 1 new case (Re* = 1) while at the
beginning of epidemic, a single case generates, on average, R0 new cases when everyone is
susceptible. This means that only a fraction, 1/R0, of the possible contacts is susceptible at
the endemic equilibrium as shown in Eq. (1-10b) (Aron J., 2007). In other words, not all
susceptible population would be infected before an epidemic stops. The larger the R0, the less
the fraction of population is left uninfected. This fact can be used to describe the herd
immunity in the SIR model. Since at endemic state, there will be certain susceptible left: S* =
N/R0. Thus, to prevent an epidemic spread out, we need to vaccinate only a fraction of the
population:
pc = (N-S*)/N = 1 – S*/N = 1 – 1 /R0
(1-11)
pc is the critical vaccination coverage fraction (Kretzschmar and Wallinga, 2010; D´ebarre,
2012).
For example, for an influenza with R0=2 (that is, one infected person will infect two people
during his/her infectious time) the fraction of the population needed to be vaccinated would
be, pc= 0.5. On the other hand, for smallpox, R0 is about 5 and pc is then around 0.8. This
means that vaccination of 50% population is enough to prevent flu epidemic, while for
smallpox, 80% population needs to be vaccinated. The fact that not everybody is needed to be
vaccinated in order to eliminate an infectious disease is known as herd immunity.
- 12 -
2. Theoretical Frame Work for Dengue Infection
2.1 Dengue transmission process
According to WHO (2012), dengue virus is transmitted to humans through the vector of
mosquitoes - the infected female mosquito bites. After a person is infected, he/she became
the main carrier and amplifying host of the virus, serving as a source of the virus for
uninfected mosquitoes for 4-5 days; maximum 12 days. Thus, mosquitoes acquire the virus
mainly from biting an infected person. Once infective, a mosquito is capable of transmitting
the virus to humans for the rest of its life through biting. In addition, an infective female
mosquito can transmit the virus to its eggs in the ovaries through vertical transmission route
as shown in Aedes albopictus mosquitoes with 75% probability (Shroyer, 1990). In contrast,
only a few percent viral transmission is through vertical route in Aedes aegypti mosquito and
the main virus transmission route is horizontal – from mosquitoes to mosquitoes through
biting infected humans.
Among the two types of dengue mosquitoes: Aedes aegypti and Aedes albopictus, the first is
the primary vector of dengue in warm climate. However, Aedes albopictus is the one that can
adapt and survive in cooler, even below freezing, temperate regions of Europe (ECDC
Technical Report, 2009).
There are four different dengue viruses - DENV-1, DENV-2, DENV-3 and DENV-4. Although
recovery from one type of virus provides lifelong immunity against the virus, it does not give
immunity against other types of viruses. In fact, “Subsequent infections by other serotypes
increase the risk of developing severe dengue…At present, the only method of controlling or
preventing dengue virus transmission is to combat the vector mosquitoes.” (WHO, 2012).
Based on this information, the modeling of dengue can be divided into two ways,
considering:
1) only one type of virus exists – infected will either die or recover and immune (SIR model
as shown in Sec. 1);
2) more than one type of virus exists - infected will either die or become susceptible again
(SIS model which will be shown next).
Other vector information is also very important, such as latency, the time delay between
being infected and becoming infectious, daily biting rate which pertains to female mosquitoes
only and strain of viruses. Since temperature influences greatly the vector’s survive, latency
and transmission capability of dengue virus, the modeling needs to consider temperature
effect for both cases. There are different ways of incorporating temperature effect to the
modeling: from average yearly temperature, to seasonal, or to daily variation. The most
important quantity that we are interested in finding out is still the basic reproduction
number in order to determine the threshold value for dengue transmission. With a vector as
the part of the transmission of dengue between humans, the process and the mathematical
expressions become more complicated compared to the one that we have discussed earlier.
2.2 A SIS model - Humans and Vectors, the Vectorial Capacity
In this modified Ross-Macdonald model, the recovered individuals become susceptible again
and there is no permanent immunity for humans (Smith et al., 2012). This is referred to as a
- 13 -
SIS model – susceptible to infected and back to susceptible again. For vectors, the infected
mosquitoes are assumed to remain infectious for the rest of their lives. Thus, only two
compartments exist for humans and for vectors: the number of susceptible - SH, SM, and the
number of infected - IH and IM, where the subscript H denotes for humans and M for
mosquitoes. In addition, for simplicity latency is not taken into account initially. This process
can be a simplified case of either dengue or malaria where the mosquito is the vector.
The flow diagram for this infectious transmission process is shown in Figure 3. Here the top
row is the transmission process for humans and the bottom is for vectors. The susceptible
humans (SH) may be moved out of its compartment through infection at the per capita rate λH
or through natural death at a per capita rate µH. The increase of susceptible humans is due to
recovery of infected humans. Here the human birth is neglected since birth rate is generally
small compared to the recovery rate. The infected humans are increased from infection of
susceptible and decreased through death due to both natural cause and disease as well as
through recovery. Similar process goes for mosquitoes except that there is no recovery for
infected mosquitoes.
death, µH, αH
death, µH
Human
vector to
human
infection
βH=mabH
Infection, λH
SH
IM
Recovery, γH
Infection, λM
IH
SM
human to
vector
infection
βM=abM
Mosquitoes
death, µM
death, µM
Fig. 3. Flow diagram of dengue infection including humans and mosquitoes.
Here λH & λM are force of infection to humans and vectors separately and are defined as
the per capita rate at which susceptible humans & vectors are infected. λH (λM) depends on
the number of mosquito bites in the human population per unit of time, NM a, where a is the
average daily biting rate per fly on humans, the infectious proportion in the female mosquito
(human) population, IM/NM (IH/NH), and the probability of viral transmission from the
mosquito (human) to human (mosquito) per bite, bH (bM). They can be expressed as:
λH = (NM a bH IM/NM)/NH
= mabH IM/NM
= βH IM/NM
(2-1)
λM = (NM abM IH/NH)/NM
= abH IH/NH
- 14 -
= βM IH/NH
(2-2)
Thus, the equivalent total transmission rate as used in SIR model from vector to the human
population is βH= mabH and from human to vector is βM= abM where m = NM/NH is the vector
(female mosquito) to human population ratio or the number of female mosquitoes per person.
Based on the flow diagram, the relevant mathematical equations are those related to infected
human and vector populations, although the other equations are also important in order to
find solutions to the whole systems:
dIH/dt = λHSH – γH IH – (µH+ αH) IH
= λHSH – (γH + µH+ αH) IH
(2-3)
dIM/dt = λMSM – µM IM
(2-4)
NH
= S H + IH
(2-5)
NM
= S M + IM
(2-6)
For an invasion of infection to take place, both infected humans and vectors must increase.
Humans: dIH/dt > 0, or λHSH – (γH + µH+ αH) IH > 0;
(2-7a)
Vectors:
(2-8a)
dIM/dt > 0, or λMSM – µM IM > 0.
Using Eqs. (2-1), (2-2), (2-5) & (2-6) and the conditions at the beginning of an epidemic:
SH = NH, SM = NM,
Eqs. (2-7a) & (2-8a) can be rewritten as:
Humans: mabH (IM/NM) NH > (γH + µH+ αH) IH; or
mabH (IM/NM) > (γH + µH+ αH) IH/ NH.
Vectors:
(2-7b)
abM (IH/NH) NM > µM IM ; or
IH/NH > µM IM / (abMNM).
(2-8b)
Combining Eqs. (2-7b) & (2-8b), we have the condition:
mabH (IM/NM) > (γH + µH+ αH) µM IM / (abMNM); or
ma > (γH + µH+ αH) µM / (abHbM);
(2-9)
ma is the mosquito biting rate per person which must meet the condition shown in Eq. (2-9)
in order for an epidemic to grow.
- 15 -
For an infection to stop, the number of both infected humans and vectors must decrease. So
the threshold conditions for epidemic to take place are when infected humans and vectors
reach constant values, or in mathematical forms:
dIH/dt = 0, & dIM/dt = 0.
The threshold condition can be rewritten as:
ma = (γH + µH+ αH) µM / (abHbM); or
mabH abM/[(γH + µH+ αH) µM] = 1.
For dengue, µH (~10-5/day) & αH (~10-3/day) are normally small and negligible relative to γH
(~10-1/day) (Massad et al., 2011). This threshold condition defines the basic reproduction
number R0 as
R0 = ma2bHbM/(γH µM)
(2-10a)
This is the relation obtained by Macdonald in his classical paper (Massad et al., 2011) on
malaria, except that the parasite latency was not incorporated here.
To incorporate latency, let n represents the pathogen (virus for dengue or parasite for
malaria) extrinsic incubation period (EIP) in days, which is the time for the vector between
being infected to becoming infective to the vertebrate host. In the experiments, it is usually
measured from the time that the mosquitoes ingested the infected human blood to the time
that the virus is found in its salivary gland or in legs and other body tissues as proxy. Also let
p be the vector’s daily survival probability. Assuming that the vector’s probability of daily
survival decreases exponentially with time, in one day
p = e-µM, or µM = - ln (p).
(2-11)
Here the per capita mortality rate of mosquito µM is measured in unit of day-1. The exponent
in p is µM (1/day) · 1 day = µM, so that there is no unit. The same goes in the expression of µM
= - ln (p) where p has no unit and implicitly µM = µM (day-1) · Time (=1 day) using day-1 as the
unit for µM. Thus, the probability of surviving the whole latency period of n days is,
pn = e-µM·n.
This is the fraction of susceptible mosquitoes that will survive the extrinsic incubation period
(n days) and become infective. Using pnSM to replace SM in Eq. (2-4) and solving for the
threshold condition as it described above, the only thing changes in Eq. (2-10a and 10b) is to
replace bH by bH pn or bH e-µM·n. Using expression for p and µM in Eq. (2-11), Eq. (2-10a) can
be rewritten as
R0 = mabHe-µM·n/µM · abM/γH
= mabHpn/[- ln (p)] · abM/γH
- 16 -
= ma2bHbMpn/[- γH ln (p)]
We can divide R0 further as consisting of two parts, one from vector to human population and
one from human to the vector population. Using the same expression R0 = β/γ from Eq. (18a) as in SIR model – the transmission rate β multiplies the infectious period (1/γ) of the
infectee, taking into account of latency R0 can be rewritten as:
R0 = R0 M->H · R0 H->M
= βH e-µM·n /µM · (βM /γH)
= mabHe-µM·n/µM · (abM/γH), or
= mabHpn/[- ln (p)] · (abM/γH).
(2-10b)
Here R0 M->H = βHe-µM·n/µM = mabHe-µM·n/µM = mabHpn/[- ln (p)], and R0 H->M = βM/γH =
abM/γH.
As discussed in Section 1, the basic reproduction number R0 represents the number of
secondary cases in the first generation produced by one primary case during his/her entire
infectious period. This applies to not only humans but also vectors. For vectors, it is during
the vector’s lifetime (1/µM) since we have assumed that the vector stays infectious for life
once it is infected and past incubation time. For humans, the infectious period is 1/γH.
Therefore, R0 M->H represents the number of humans infected by one infectious mosquito
during its lifetime after being introduced to an entirely susceptible human population.
Similarly, R0 H->M represents the number of infected mosquitoes produced by one infectious
human during his/her infectious period after being introduced to fully susceptible vector
population. Since there is no recovery in mosquitoes, the vector population are assumed to be
susceptible. Combining these two parts, the basic reproduction number means that the
number of new human cases generated by one infective human during his/her infectious
time (about 10 days for dengue) after being introduced to a fully susceptible human
population, through vectors who have survived incubation time after being infected by the
infective human.
As before, R0 > 1 is the condition for an epidemic to take place. This condition does not
require either R0 M->H or R0 H->M to be larger than 1 but the product of them. Thus, if the
transmission rate from human to vector (βM) is low or the recovery rate (γH) is high so that
R0H->M < 1, the epidemic can still take place if the transmission rate from vector to human
(βH) is high due to a large vector to human ratio or a low vector mortality rate so that R0 M->H
>1.
Another quantity called Vectorial Capacity, V, is often used to characterize the vector’s
ability in transmission disease. It is defined as
V = R0·γH
= R0 M->H ·R0 H->M · γH
- 17 -
= ma2bHbM e- µM·n /µM, or
= ma2bHbMpn/[-ln (p)]
(2-12)
From this equation, we may say that only the second part of R0, R0 H->M, is averaged by the
duration of infectiousness of the infected person – D=1/γH , that is, R0 H->M/D. The first part
is intact, not averaged over vector’s lifetime. This means that V represents the number of
infected vectors per unit time, e.g. a day, when an infected person is introduced to a
susceptible vector population multiplies the number of new cases in humans produced
during each infected vector’s life time after being introduced to a fully susceptible human
population. In other words, the vectorial capacity is the average new cases generated per
unit time by one infected case introduced in a fully susceptible population during his/her
infectious period.
As suggested by Garrett-Jones & Grab (1964, p83) using day as the unit of time, the vectorial
capacity represents the average daily number of secondary cases generated by one primary
case introduced in a fully susceptible host. Hence, he called the term "vectorial capacity” the
"daily reproduction rate, that is, the daily fraction of the basic reproduction rate.” Here
the basic reproduction rate refers to the basic reproduction number R0 used in this thesis.
Thus, the vectorial capacity can be called also the basic daily reproduction number.
From Eq. (2-10b), it is possible to estimate the critical density of female mosquitoes (number
of female mosquito per host), m*, below which the disease will naturally disappear by setting
R0 = 1:
m* = - γH ln (p) /a2bHbMpn
(2-13)
2.3 Effects of Temperature in Dengue – Seasonality
2.3.1 Sinusoidal Variation of Transmission Rate
Temperature affects the behavior of vectors: its population NM, biting rate a, biting capacity –
bV and bH, incubation time n, daily survival probability p or mortality rate µM, and eggs
hatching rate, etc. Thus, temperature affects the basic reproduction number or the vectorial
capacity. The temperature effect has been seen in the dengue cases where they increase
during summer and disappear during winter. As climate changes over the last few decades,
the ambient temperature increase has possibly contributed to the drastic increase of the
dengue cases, such as in Singapore – a more than 10-fold increase between 1989 and 2005
(Massad et al., 2011). Thus, both average temperature and its variation are important factors
to be taken into account in modeling.
A simple way to incorporate the seasonal change of temperature may be to reconsider the SIR
model of section 1. The temperature effect on vector’s behavior may be simplified and
reflected on the disease transmission rate, β(t), as a time dependent sinusoidal function with
a period of one year (Coutinho et al., 2006).
β(t) = β[(T − A sin(2π f t + φ)] · θ[T − A sin(2π f t + φ)],
- 18 -
(2-14)
where T, A, f, and φ are constant parameters to represent the temperature with varying
amplitude A around a constant temperature T, and the frequency f which is 1/365 days-1 and
phase φ to set initial transmission rate. θ(x) is the Heaviside function with a value of 1 when x
is positive and 0 when x is negative, to guarantee that β(t) stays positive.
The threshold condition (Re=1) as expressed by the effective reproduction number, Re, in Eq.
(1-7) is now also a periodic function of time,
Re(t) = β(t)S(t)/(N(t)γ)
As R(t) > 1, the number of infected individuals (I) increases after a time delay. As Re(t) < 1,
the number of infected individuals decrease. Thus, as Re(t) varies periodically from below 1 to
over 1 within one year period, the infected individuals also varies periodically as shown in Fig.
4 based on Eqs. (1-6) and (2-14), a simulation done by Coutinho et al. (2006). In addition, the
maxima of I increase as the average (over one year) Re-avg(t) > 1 for the first 11 years and
decrease as the average Re-avg(t) < 1 after that.
Fig. 4. Simulation of a system with sinusoidal transmission rate (Coutinho et al., 2006). The time unit is in days.
2.3.2 A Modified SIR model – Humans, Vectors and Eggs
A more advanced Ross-Macdonald model to incorporate seasonal temperature variation is
given by Massad et al. (2011) & Coutinho et al. (2006). In this model, they have considered
three components in the dengue transmission process: humans, mosquitoes and their eggs
which include other intermediate stages like larvae and pupae. It is the mosquito eggs who
survive the winter. This component is necessary in the model in order to explain the observed
dengue phenomena: overwintering without assuming unreasonably high biting rate
(Coutinho et al., 2006).
The flow diagram is shown below which is created based on the mathematical equations
given by Massad et al. (2011) & Coutinho et al. (2006).
The relevant equations are those describing the changes of Infected humans (IH), Infected
female mosquitoes (IM), Infected eggs (IE), and latent mosquitoes (LM), although to solve the
equations, a full set of 9 differential equations are needed – one for each compartment and
- 19 -
one for the total human population which is not constant (Coutinho et al. (2006; Massad et
al., 2011).
dIH/dt = λHSH – (γH + µH+ αH) IH
(2-15)
dLM/dt = λMSM – e - µM·τ λM(t-τ)SM(t-τ) – µM IM
(2-16)
dIM/dt = e - µM·τ λM(t-τ)SM(t-τ) – µM IM + pI ·FI t)
(2-17)
dIE/dt = grM IM – µE IE – pI ·FI(t)
(2-18)
death, µH, αH
death, µH
Human
Infection
λH
SH
vector to
human
infection
βH=mabH
Infection
e
IM
- µM·τ
λM(t-τ)
IH
Recovery
γH
Infection
λM
Latent
LM
lay egg
rM
Mosquitoes
µM
lay egg grM
Eggs
egg hatch
pI·FI (t)
IE
µM
SE
(1-g)rM
death, µE
death, µH
RH
human to
vector
infection
βM=abM
SM
µM
egg hatch
ps·Fs t)
death, µE
Fig. 5 Flow diagram of dengue infection including humans, mosquitoes and eggs.
Here λH = abH IM/NH (= mabH IM/NM) and λM = abH IH/NH are the same as in Eqs. (2-1) & (22). τ is the Latent period (n) which is used to express the time delay from infection to
transmission. Term e- µM·τ expresses the fraction of mosquitoes that survive the latency
period. pI is the per capita egg hatch rate, and
FI(t) = [(T − A sin(2π f t + φ)] · θ[T − A sin(2π f t + φ)].
(2-19)
FI(t) is the seasonal variation in mosquito production from infected eggs. g is the proportion
of infected eggs laid by infected female mosquitoes. rM is the oviposition rate of eggs which
depends on the density of eggs due to limited breading places:
rM = rM0 [1- (SE + IE)/kE]
(2-20)
Here, rM0 and kE are constants.
- 20 -
As shown by Massad et al. (2011), to solve the approximated threshold condition we can
assume that some disease is introduced at time = 0 and then ‘freeze’ the system at time t. The
effect of a small amount of infected cases introduced on the stability of the frozen equilibrium
is investigated - solving the four differential equations (2-15) to (2-18). The threshold
condition is obtained if the effective reproduction number meets the criterion:
Re(t) = abM abH SM(t − τ)·exp (−μMτ)·SH(t)/[(γH + αH + μH) μM NH(t − τ)·NH(t)]
+ pI ·FI(t) g rM/ [μM· (μE + FI(t))]
(2-21)
If we look at the initial condition (t=0) for a whole susceptible population: SM(t − τ) =NM,
SH(t) = NH, αH & μH << γH, the first term is the same as the basic reproduction number R 0
expressed in Eq. (2-10b). The second term takes into account the effect of seasonal variation
in temperature on the mosquito behavior through the egg population. Thus, when Re(t) > 1,
the system is unstable and the epidemic takes off. The basic reproduction number alone (the
first term) is not sufficient in determining the threshold of epidemic after a small amount of
disease is introduced at time t = 0.
With this seasonal dependent reproduction number Re(t), Coutinho et al. (2006) has
successfully explained the dengue overwintering and the observed delay of a few months
between mosquitoes population density and the peak in dengue cases. Their simulation
showed that the vertical transmission through eggs is more likely responsible for annual
returning sustained transmission than particularly long-lived female mosquitoes for
overwintering (and dry season). This model is useful and relevant to our interest in dengue
suitability in Europe where the Aedes albopictus mosquitoes is the main concern for its
survival ability in cooler European climate and its high vertical virus transmission probability
through eggs (Shroyer, 1990).
Here the work presented for dengue has taken into account only seasonal changes of
temperature. The seasonal temperature effect is accounted through the change of egg
population only. All other adult mosquito behavior parameters in Re have been assumed
constant in this model by Coutinho et al. (2006) & Massad et al. (2011).
2.4 Effects of Temperature in Dengue – Daily Fluctuation
Recent empirical study by Lambrechts et al. (2011) on Aedes aegypti mosquito showed that
daily temperature variation also had strong effect on the mosquito’s behavior. They did both
experiments and computer simulations using thermodynamic modeling.
The dependence of diurnal temperature range - DTR or daily temperature variation
amplitude - on four parameters were examinated: the probability of infection from human to
vector - bM, the probability of transmission from vector to human - bH, Extrisic Incubation
Period (EIP) - n, and the survival probability, Sv(t) which is related to the vector mortality
rate - µM. In their laboratory experiments, Lambrechts et al. (2011) have varied the daily
temperature around an average of 26 0C with 3 amplitudes (diurnal temperature range DTR): 0, 10 and 20 0C where a sinusoidal rising and exponential decay are used to imitate
observed daily temperature fluctuation (12 hours above mean and 12 hours below mean
temperature) in Thailand during high (DTR=10 0C, summer) and low (DTR=20 0C, winter)
- 21 -
season of dengue. After feeding the female mosquitoes with infected blood from Thailand of
virus DENV-1 and DENV-2 separately, they measured bM by checking the prevalence of
midgut infection of the vector, and viral dissemination from midgut to other body tissue
which is equivalent to bH, Sv(t) the probability of vector survival by counting how many died
every 1-3 days, and n the pathogen EIP duration by the time required for viral dissemination
from the midgut to other tissues.
Their results showed that the average infected female mosquitoes over the entire time course
(32 days) were 97.1, 94.9 and 78.9% for DENV-2 under DTRs of 0, 10 and 20 0C, and 97.0
and 88.4% for DENV-1 under DTRs of 0 and 20 0C. Increasing DTR reduces the likelihood
that a female became infected (bM). This finding is confirmed by their simulation using
thermodynamic model to investigate the same effect (bM) over a wider range of temperature
(mean temperature 14 ≤ T ≤ 28 0C and 0 ≤ DTR ≤ 22 0C). They also found that at a mean
temperature lower than about 19 0C, large DTRs increase infection probability, while in
environments higher than 19 0C, large DTRs reduce infection probability. From the 3-d map
of the dependence of bM on T and DTR, I have extrapolated bM values for two mean
temperatures: T= 14 and 26 0C which is shown in Figure 6.
Fig. 6. Vector infection and transmission probabilities as a function of DTR based on simulation result (Lambrechts et al.,
2011).
The same simulation on the transmission probability (related to bH) has shown a stronger
nonlinear response to temperature than infection probability (bM). Similar to infection
probability, the probability of transmission showed also sign reverse depending on the mean
temperature. At a mean temperature lower than 18 0C, the large the DTR is, the more
increased the transmission probability is; while above 18 0C, the large the DTR is, the more
reduced the transmission probability is. At mean temperature T=14 °C, a 2.7-fold increase
(from 0.11 to 0.297) is found for transmission probability (bH) and 2.4-fold increase is found
for infection probability (bM) as DTR increases from 0 to 20 °C. However, at T=26 °C, the
temperature fluctuation DTR is more dramatic to bH than to bM: as DTR increases from 0 to
20 °C, a 2.1-fold decrease is found for transmission probability (bH, from 0.95 to 0.45) and
only 1.3-fold decrease for infection probability (bM). Figure 6 shows the simulation results for
two mean temperatures discussed here. However, this DTR dependence of bH was not found
in their experiment from the measurement of the prevalence of dissemination (bH). Here bH
is the probability of infection to human per bite. The experiment measures the prevalence of
dissemination which is related but not exactly the same as bH. This may account for the
differences between measurement and simulation.
- 22 -
In their experiments on survival analysis, a significant difference was observed of DTR on
overall survival probability, Sv, although mean survival time (27.3-27.9 days) was similar
across different DTRs for DENV-2. At the end of experiments (32 days), 30% females under
DTR of 20 °C survived compared to 50 and 70% at DTR of 10 and 0 °C for DENV-2. Whereas
for DENV-1 virus, about 10% vs. 60% survived in the end at DTR of 20 and 0 °C, and the
mean survival times were different and corresponded to 22.2 and 28.5 days. No effect of
DTRs on EIP or n, was observed in their experiment. From their simulation a decreasing
function of n on mean temperature is found which is supported by other studies (Gubler,
1998).
- 23 -
Results
1. Combined Effects of Temperature on Dengue Relative Vectorial
Capacity
So far, we have seen a description of two Ross-Macdonald mathematical models (SIS and
modified SIR) on dengue transmission process, and two examples of temperature effects –
seasonal and daily fluctuation - on reproduction number and on some of the dengue vector
parameters. In each of these two examples the model is limited since each considered part of
the temperature effect and neglected the other part. This section will focus on a combined
effect of temperature on dengue suitability by using the results of mathematical modeling
presented in Methods Section 2.2 & 2.4 and connecting the data from empirical studies both
in Section 2.4 and other literature. Without using a computer program to simulate the
temperature (seasonality) change on dengue through mosquito’s egg hatching behavior, the
focus here is on how each parameter depends on temperature and their effect on the dengue
transmission. The seasonality can be viewed as being built in the effect of temperature on
adult female mosquito’s parameters in both its mean value – low in winter and high in
summer, and in daily fluctuation range which has shown large in winters and small in
summers in sub-tropical regions, such as, Thailand (Lambrechts et al., 2011).
To predict the start of an epidemic, the vectorial capacity, V, is often used for vector-borne
infections instead of the basic reproduction number R0. They differ by the human recovery
rate from the infection that is normally temperature independent. Thus, vectorial capacity
depends on temperature the same way as the basic reproduction number but involving only
vector parameters. As expressed in Eq. (2-12), vectorial capacity depends on, m, the vector to
human population ratio, or the number of female mosquitoes per person, which varies from
location to location. To get a general idea without limiting to a specific location, the relative
vectorial capacity, V/m, will be used in the estimation of dengue risk based on temperature.
The expression of Eq. (2-12) can be rewritten as:
V/m = a2bHbM e-µM·n/µM
(2-22)
In principle, every parameter here depends on temperature and particular species of
mosquito and virus. The mostly studied dengue mosquito is Aedes aegypti whose data will be
used here, although ideally both dengue mosquitoes, Aedes aegypti and Aedes albopictus,
need to be parameterized and modeled. There is no distinction between different viruses due
to lacking of data.
1.1 Effect of Mean Temperature on Dengue Relative Vectorial Capacity
Through literature search, I have found and listed below the dependence of all five vector
parameters, a, bH, bM, n, and µM on temperature for Aedes aegypti mosquito, where T is
either mean or ambient temperature depending on studies.
1) Biting rate a
- 24 -
According to experimental study by Scott et al. (2000) in Thailand, the blood feeding
frequency of female Aedes aegypti collected weekly in Thailand (1990-1992) showed the
following linear regression fitting:
a = 0.03T + 0.66.
(2-23)
The blood feeding increases significantly (p=0.05) with temperature for the range 21 °C ≤ T ≤
32 0C measured and the unit is 1/week. From this relation, the average daily biting rate a
(dividing the equation by 7) increases slowly with T at the values from 0.18/day at T = 21 °C
to 0.23/day at T = 32 0C. They showed many measurement data. In addition, different
locations, e.g. Brazil, showed different linear fitting – independent of temperature. Given this
fact and the variation of a on temperature is small across the measured range, we may extend
the relation to lower temperature. For example, at T = 12.4 and 14 °C, a is calculated to be
0.147 and 0.154/day based on Eq. (2-23), where the values of a at these low temperatures are
not very different from the experiment data observed on higher temperatures, e.g. 0.18/day
at T = 21 °C.
2) The probability of infection from human to vector per bite - bM
Based on empirical data from many studies in the range of 12.4 °C ≤ T ≤ 32.5 °C, Lambrechts
et al. (2011) modeled both infection and transmission probabilities and found the following
relation from fitting the data: in the range of 12.4 °C ≤ T ≤ 26.1 °C, bM increases linearly with
T as
bM =0.0729T −0.9037
(2-24)
until it reached one and then remained equal to 1 for 26.1 < T < 32.5 °C.
The empirical data that they used were chosen from different studies that measured the
highest proportion of infected and transmitting vectors at various constant temperatures and
for the flaviviruses used: West Nile virus, Murray Valley encephalitis virus, and St. Louis
encephalitis virus.
3) The probability of infection from vector to human per bite - bH
From the same study as above, Lambrechts et al. (2011) modeled and found a nonlinear
relation of bH on T: bH increases almost linearly with T for 12.4 ≤ T < 26 0C, decreases
sharply when T> 28 0C and to zero at T =32.5 0C. The fitting relationship between bH and T is
described by the following equation:
bH = 0.001044T (T – 12.286) (32.461 - T)1/2.
(2-25)
4) Extrinsic incubation period - n
Time required by DENV and many other mosquito-borne viruses to complete extrinsic
incubation in the vector is temperature sensitive (Watts et al., 1987). Using an enzyme
kinetics model to fit experimental data, Focks et al. (1995) showed a decreasing relationship:
from about 40 to 5.4 days as T increases from 12 to 36 °C. The simulation is based on
- 25 -
experimental data range from 12 ≤ T ≤ 35 °C. Since there is no analytical equation given, I
used an exponential model to fit their data and obtained the following relation:
n = 4+e(5.15-0.123T).
(2-26)
Since the temperature range is less than one order of magnitude, the fitting relation Eq. (226) with 3 constants is not unique. Other relations may work also, such as, polynomial. The
exponential function was chosen because it has been used in other modeling for n in malaria
mosquitoes (Massad et al., 2011). Within the range of temperature used in this estimation it
is likely not so affected by the model assumptions,
5) Mortality rate - µM
The per capita female Aedes aegypti mosquito’s mortality rate (e.g. the incidence rate) was
studied by Yang et al. (2009). Their experimental data measured in the temperature range of
10.54 °C ≤ T ≤ 33.4 °C showed that the mortality rate µM ranged from 0.027 to 0.092/day.
The mortality rate is the highest at both low (T < 14 °C) and high temperature (T > 32 °C) and
remains nearly constant in between. The lowest mortality rate 0.27/day occurs at T = 27.6 °C.
To fit the data, they found a 4th order polynomial function:
µM = 0.8692 – 0.1590 T + 0.01116 T2 – 3.408x10-4 T3 + 3.809x10-6 T4.
(2-27)
When putting all the analytical relations in the spread sheet, the dependence of the five
parameters mentioned above on temperature is calculated. Their dependence on mean or
ambient temperature is shown in Figure 7.
Fig. 7. Vector parameters for Aegypti as a function of mean temperature.
(Notice, different scales are used to bring all the parameters under one graph.)
- 26 -
From these data input, the relative vectorial capacity, V/m, is then calculated using Eq. (222). The result of V/m as a function mean or ambient temperature is shown in Fig. 8. A
nonlinear relation is observed. When temperature is in the range 12.4 ≤ T < 28 0C, a
monotonic increasing dependence on temperature is observed and when T> 28 0C up to 32.5
0C, a sharp decreasing relation is found. Thus, at both high and low ends of the mean
temperatures, the vector Aedes aegypti mosquito’s dengue transmission capability is
reduced. While around 28-30 0C, the relative vectorial capacity reaches its peak. Thus, 28-30
0C is the optimal mean temperature for Aedes aegypti mosquito to transmit dengue. At
temperature T = 14 0C, the vectorial capacity is reduced by 770 times from its peak value
around 28-30 0C. As mean temperatures reduces under 26 0C, all the five vector parameters
contribute to the reduction of the relative vector capacity; while at high mean temperatures
(over 30 0C), only two vector parameters contribute to the reduction of the relative vector
capacity - the mosquitoes’ mortality rate and the probability of transmission from vector to
human.
1.4
V/m (1/days)
1.2
1
0.8
0.6
0.4
0.2
0
12
16
20
24
T (ºC)
28
32
Fig. 8. Relative vectorial capacity as a function of mean temperature.
1.2 Effect of Daily Temperature Variation on Dengue Relative Vectorial
Capacity
The temperature results in Fig. 8 were obtained without considering the daily temperature
variation. Here, we will incorporate diurnal temperature range – DTR in the vectorial
capacity calculation. However, the work in this area on vector parameters is scarce and the
work by Lambrechts et al. (2011) seems the only available study of dengue on DTR.
As shown in Methods section, Lambrechts et al., (2011) found in their experiments and at T =
26 0C that DTR had strong influence on the two vector parameters: the probability of
infection per bite to vectors (bM) and survival probability Sv(t) which is related to mortality
rate – see below. In addition, they have simulated two parameters, bH and bM, as a function of
mean temperature and DTR in a 2-d map from 14 0C ≤ T ≤ 28 0C & 0 0C ≤ DTR ≤ 22 0C.
Using their simulation map, the dependence of bM and bH on DTR can be extrapolated for
each mean temperature. Two of the DTR dependent relations at mean temperatures of 14 and
26 0C were shown in Figure 6. These data will be used as examples to show the DTR influence
on V/m at these two mean temperatures.
- 27 -
In the measurement of DTR dependence on EIP– n, Lambrechts et al. found no variation
over DTR from 0 to 20 0C. From temperature dependent relations Eq. (2-26), the extrinsic
incubation period (n) is found to be 34 days at T=14 0C and 12 days at T=26 0C. This result –
12 days - is similar to their and others’ experimental findings of 8-14 days at T=26 - 30 0C
(Gubler, 1998).
In the survival analysis at T = 26 0C, as mentioned in Methods section 2.4, Lambrechts et al.,
(2011) found quite different values for the survival probability, Sv(t), at the end of experiment
(duration of 32 days) depending on the DTRs and virus types, see Table 1. From this data, the
daily average mortality rate, µM, can be estimated based on the definition through the
following equation:
µM = Ndead/(Ntotal*D)
= (Ntotal - Nsurvival) /(Ntotal*D)
= (1 - Sv)/D.
(2-28)
Here Ndead is the number of dead mosquitoes during the observation time - D = 32 days and
Ntotal is the total mosquitoes observed where Ntotal = Ndead + Nsurvival. The survival probability,
Sv = Nsurvival/Ntotal.
For the two types of viruses studied, DENV-2 and DENV-1, the calculated vector daily
mortality rates for Aedes aegypti mosquito at T = 26 0C are listed in Table 1.
Table 1. Vector mortality rate (µM) and survival probability (Sv) measured for Aedes aegypti mosquito at mean temperature
0
T = 26 C and different diurnal temperature range (DTR) (Lambrechts et al., 2011).
DENV-2
0
DENV-1
-1
-1
DTR ( C)
Sv (%)
µM (days )
Sv (%)
µM (days )
0
70
0.00938
60
0.0125
10
50
0.01563
20
30
0.02188
10
0.02813
Fig. 9. Vector mortality rate as a function of DTR for two dengue viruses based on data in Table 1.
- 28 -
As shown in Figure 9, the vector mortality rate increases linearly as the daily temperature
fluctuation increases for both types of viruses. At mean temperature T = 26 0C, the larger the
daily temperature fluctuates, the less likely the mosquito is going to survive.
The value at highest DTR is very close to the experimental study by Yang et al. (2009), where
the mortality rate is 0.03/day at T = 26 0C and 0.04/day at T = 14 0C as obtained from their
fitting to the measured data for Aedes aegypti mosquitoes in Brazil. As mentioned in Result
section 1.1, their range of measurement for mean temperature is from 10.5 to 33.4 0C at a
constant DTR for each T. Specifically, a fixed two-step temperature per day is used to
simulate natural photo period for day and night. For example, at mean temperature T = 15 0C,
the daily temperature is 17.9 0C for 10 hours and 12.1 0C for 14 hours to imitate winter; at T =
25 0C, the daily temperature is 27.6 0C for 13 hours and 22.4 0C for 11 hours to imitate
Spring/Fall. Thus, a constant value of µM = 0.04/day will be used for T = 14 0C to calculate
V/m.
Due to lacking of data, a constant average daily biting rate a - independent of DTR - will be
used. From temperature dependent relations Eq. (2-23), the value of a is 0.21/day at 26 0C
and is 0.15/day at T=14 0C assuming that it is still valid at this temperature (Scott et al.,
2000). These values are similar to that used by Massad et al. (2012) for dengue in Thailand 0.164/day, but less than that for dengue in Singapore – 1.2/day (Massad et al., 2011).
0.014
4
0.012
3.5
3
0.01
T = 14 ºC
V/m (1/day)
V/m (1/day)
When putting all the values of the parameters mentioned here in Eq. (2-22) where mortality
rate for DENV-2 is used at T = 26 0C, V/m is calculated as a function of daily temperature
fluctuation – diurnal temperature range (DTR). At mean temperature T = 14 and 26 0C, the
results of V/m are shown in Figure 10.
0.008
0.006
0.004
T = 26 ºC
2.5
2
1.5
1
0.002
0.5
0
0
0
5
10
15
Daily temperature variation DTR (ºC)
20
0
5
10
15
Daily temperature variation DTR (ºC)
0
20
0
Fig. 10. Relative vectorial capacity as a function of DTR at two mean temperatures, 14 C (left) and 26 C (right).
A reversed dependence of relative vectorial capacity on DTR is observed in Figure 10 at the
two mean temperatures. At T=14 0C, large daily temperature fluctuation amplitude, will
enhance the vector’s ability to transmit dengue infection between humans. For example, at
DTR=20 0C which corresponds to that temperature changes daily from 4 to 24 0C, 6.4 times
increase is expected when comparing to a constant daily temperature of 14 0C. In contrast, at
high temperature T=26 0C, the vector’s ability to transmit dengue decreases with the range of
temperature fluctuation. For example, at DTR=20 0C which means that daily temperature
changes from 16 to 36 0C, V/m decreases by a factor of 7.3 relative to the constant of 26 0C
daily temperature. The exact reason for this temperature behavior is unclear. The suspected
- 29 -
reason may be “deleterious effects of low and/or high temperatures on key steps of the
progression of virus infection in the mosquito (Lambrechts et al., 2011).”
In addition, the ratio of V/m between the two mean temperatures (V/m at T=26 0C over V/m
at T=14 0C) showed a strong dependence on DTR - from 1881 at DTR=0 0C to 40 at DTR=20
0C. Thus, in a place with mild summer (T=14 0C) but large daily temperature variation such
as in Sweden, the dengue transmission may not be as small as it would be if the daily
temperature variation is small.
At T=26 0C the value of relative vectorial capacity obtained in Fig. 8 without considering
temperature fluctuations specifically is V/m = 0.95/day, which is a factor of 1.8 greater than
that at DTR=20 0C and 4 times less than that at DTR=0 0C. It corresponds to the DTR around
14 0C.
2. Sensitivity Analysis of Vector Parameters on Dengue Vectorial
Capacity
Since different values have been used for Dengue in the literature for the same parameter in
calculating vectorial capacity (V) or basic reproduction number (R0), it would be interesting
to see how much those values affect the result. Thus, a sensitivity analysis is performed here
and illustrated by some commonly used values for the parameters.
The sensitivity of V/m can be shown mathematically through partial derivative, ∂(V/m)/∂x to
each parameter x, in Eq. (2-22), assuming that other parameters are constant (Massad et al.,
2011). Let V*= V/m to simplify the expression.
∂V*/∂a
= 2abHbM e-µM·n/µM = 2V*/a
(2-29a)
∂V*/∂(bH)
= a2bMe-µM·n/µM = V*/bH
(2-30a)
∂V*/∂(bM)
= a2bHe-µM·n/µM = V*/bM
(2-31a)
∂V*/∂n
= - µM a2bHbM e-µM·n/µM = - µM V*
(2-32a)
∂V*/∂µM
= - n a2bHbM e-µM·n/µM - a2bHbM e-µM·n/µM2 = - (n+µM-1)V*
(2-33a)
From these equations, it is clear that parameters a, bH, bM affect V* positively, n and µM affect
V* negatively. In other words, as either a, bH or bM increases, V* increases as well. However,
as either n or µM increases, V* decreases as the negative signs in Eqs. (2-32a & 33a) show.
Here, the magnitude of change in V* relative to the change of each parameter can be
compared by re-arranging Eqs. (2-29a) to (2-33a):
∂V*/V* (relative to a) = 2 ∂a/a
(2-29b)
∂V*/V* (relative to bH) = ∂bH/bH
(2-30b)
∂V*/V* (relative to bM) = ∂bM/bM
(2-31b)
- 30 -
∂V*/V* (relative to n) = - µM ∂n = - µMn ∂n/n
(2-32b)
∂V*/V* (relative to µM) = - (n+µM-1) ∂µM = - (µMn +1) ∂µM/µM
(2-33b)
If the relative change of each parameter ∂x/x is 1%, the effect on V* - the relative change of
vectorial capacity, ∂V*/V* (= ∂V/V), would be those values as shown in Table 2.
Table 2. Comparison of relative change in vectorial capacity V* (= V/m) as each vector parameter varies 1%.
Relative change of parameter
∂a/a = 1%
∂b /b = 1%
∂b /b = 1%
∂n/n = 1%
∂µM/µM = 1%
∂V*/V* (%)
2
1
1
- µM n
-(µMn +1)
H
H
M
M
Since n is on the order of 10 or more, µM n >1 if µM is larger than 0.1/day. However, for the
values used for plotting Figure 8 & 10 (Table 1), µM n < 1. Thus, the magnitude of relative
change in V* is the largest when the biting rate a changes. In other words, the biting rate a is
the most sensitive parameter in determining vectorial capacity providing µM n < 1. The next
sensitive parameter is the mortality rate µM. The least sensitive parameter is the EIP – n.
This ranking is based on the theoretical assumption that each parameter can change
relatively the same amount and µM n < 1. However, in reality the range of change for each
parameter varies and this range is just as important as the theoretical analysis. Let us see
some of the examples.
1) Daily biting rate a
Figure 8 & 10 used the calculated values of vector parameters based on temperature relation
obtained from experimental study (Scott et al., 2000). If varying a from 0.206/day – the
value measured by Scott et al. (2000) at T=26 0C - to 0.164/day as used by Massad et al.
(2012) for dengue study in Thailand – 1.25-fold reduction in a, the V/m decreases by a factor
of 1.57 throughout the whole DTR range. The same way of varying a at T=14 0C from 0.154 to
0.164/day – 1.06-fold increase in a, V/m increases 1.13 times for the whole DTR range. A
square depending relation is confirmed. Thus, an accurate measurement of a is important in
principle but not so critical in reality on the value of vectorial capacity due to both the its less
sensitive dependence on temperature and the relative small range of value used in the
different modeling of dengue.
2) Extrinsic Incubation Period – n
Different values of the extrinsic incubation period (EIP) n for dengue vector has been
reported and used in modeling in the literature based on different studies. Although the
general reported values are between 8-14 days for mean temperature T= 30 to 26 0C, n = 7
days has been used also for dengue study in Singapore (Massad et al., 2011), where the mean
temperature is within this range – from 27.3 to 28.1 0C during 1989 to 2005. These range of
values will be used here to show the comparison of n= 7 - 14 days on V/m at mean
temperature of 26 0C. n= 12 days is the one used in Figure 8 & 10. As n increases by a factor
- 31 -
of 2, V/m decreases from 7% to 17% depending on the value of DTR. As expected from
theoretical analysis shown in Table 2, the relative change of n has little effect on relative
change of V* since a small mortality rate (see Table 1) is used in this case. However, if a large
mortality rate, such as, µM = 0.263/day (instead of 0.01 - 0.02/day as used in Fig. 10) would
be used as discussed in the next section, a relative larger effect of 6.3 time decrease on V*
would result. However, this is still small considering a factor of 13-26 fold increase in µM.
V/m (1/days)
At T= 14 0C, the n used in Figure 8 & 10 is 34 days. If n varies 10 days in each direction – a
factor of 1.8 time increase from the lowest to the highest value, the effect on V/m is a factor of
2.2 reductions as shown in Figure 11 where mortality rate is fixed at µM =0.04/day. The effect
of n at T= 14 0C is much bigger than that at T= 26 0C. This is because T= 14 0C, µM n= 1.36
which is larger than 1 and much larger than the values of µM n = 0.12 – 0.24 at T= 26 0C (the
variation is for the range of DTR). Thus, as expected from the theoretical analysis in Table 2,
n is not a very sensitive parameter to the vectorial capacity at low µM. Its sensitivity increases
as µM increases and needs to be considered as µM n > 1.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
n=24 days
n=34 days
n=44 days
0
T = 14 ºC
5
10
15
Daily temperature variation DTR (ºC)
20
0
0
Fig.11. The effect of different Extrinsic Incubation Period n on relative vectorial capacity, V/m. Left: T= 14 C. Right: T= 26 C
µM varies with DTR as shown in Table 1.
3) Mosquito mortality rate µM
In the mathematical modeling studies, a constant value µM = 0.263/day were used by Massad
et al. (2011; 2012) for dengue studies in both Thailand and Singapore. Thus, the value of
µM=0.263/day will be used here to compare with the values used in Figure 10. Figure 12
shows the dependence of V/m on µM for the two mean temperatures. At T=14 0C, as the per
capita vector mortality rate µM changes between the two constant values: from 0.04/day to
0.263/day – a factor of 6.6 times increase, about 13,000 - fold decrease is found in V/m!
At T=26 0C, as µM changes from a range of values: 0.01 to 0.02/day (Table 1 DENV-2) to a
constant value of 0.263/day – a factor of 26 to 13 times increase, V/m decrease by a factor
from 588 to 217 times as DTR changes from 0 to 20 0C. Thus, the vector mortality rate is the
most sensitive vector parameter affecting the value of vectorial capacity. This is because that
V depends on µM exponentially. At small value of µM and n (e.g. at T= 26 0C), it is small
change on V but at large value of µM and n, the effect is quite big as shown here at T=14 0C as
well as illustrated in table 2. Thus, to identify the proper value of µM and its dependence on
- 32 -
temperature is the most important step to take for improving the accuracy of modeling on
predicting the dengue risk.
Fig. 12. The effect of different mortality rate on relative vectorial capacity. Varying µM means that it varies according to
daily survival probability as shown in table 1.
- 33 -
Discussion
We have seen in the Results section that the relative vectorial capacity, V/m, depends on
temperature, both its mean value T and its daily fluctuation amplitude – DTR through the
five vector parameters as expressed in Eq. (2-22):
V/m = a2bHbM e-µM·n/µM
As mean temperature increases, the vectorial capacity increases slowly at low temperature
and speeds up as temperatures approaches 26 0C. After reaching its peak around 28-30 0C,
the vectorial capacity decreases sharply. The ratio of V/m at peak T = 28 0C relative to 12.40C
is over 7 orders of magnitude. Thus, practically speaking, there is little dengue risk when T <
14 0C, based on the result of this study from combined temperature effect.
As DTR varies, the calculated result showed that V/m increases with DTR at low mean
temperature T=14 0C and decreases with DTR at high mean temperature T=26 0C. The value
of V/m at T=26 0C is about 1881 times greater than that at T = 14 0C when there is no daily
temperature fluctuation. However, at DTR = 20 0C this ratio is decreased to about 40. Thus,
large daily temperature fluctuation increases the dengue transmission at low mean
temperatures and reduces the gap between warmer and cooler region drastically, e.g. over 46
fold as DTR increases from 0 to 20 0C between two areas with mean temperatures of 14 and
26 0C.
This result is important considering the ongoing climate change, which comes with large
temperature fluctuation in additional to overall temperature increase. These two mean
temperatures (14 and 26 0C) used in the example could describe two different seasons in one
place, for example the spring or fall and summer in a city in Southern Europe, or two
different places in the same season of summer, such as, one in Thailand and one in Europe.
Thus, the cooler region may have higher risk than expected based on warmer region’s data,
such as, Thailand where the typical daily temperature fluctuation is around 10 0C. This daily
temperature fluctuation is not as large as in temperate region of Europe, such as, Sweden.
This result also helps us to understand when a place in Europe may be at risk of dengue
epidemics, e.g. the time of the year and how long this risk period potentially may be,
according to the climate data.
These results reflect the dengue vector, Aedes aegypti’s behavior in dengue transmission,
given in the parameters measured for carrying certain type and amount of virus. These
parameters depend on the biological condition and physical environment experienced in the
experiment or data collection in the nature. However, dengue transmission seems also to
depend on where the virus is from and which sub species are used as viruses also change over
time (Hii, 2012). Thus, we need to be careful in generalizing this result to specific places. The
local temperature and proper vector and pathogen data need to be used in the model.
Sensitivity analysis indicates that not all the parameters affect vectorial capacity the same
way. The most sensitive vector parameter is the mortality rate especially when it is larger
than 0.1/day. Small variation in mortality rate makes a large difference in vectorial capacity,
especially if combined with longer extrinsic incubation time, n. Vector biting rate is in
- 34 -
principle a sensitive parameter but in reality it seems not varying so much with temperature.
The less sensitive parameter is extrinsic incubation period, n, especially its direct effect to
relative vectorial capacity at high mean temperatures. However, it amplifies the effect of
mortality rate through the vector’s survive probability of EIP, e-µM·n. At low temperatures
where both vector’s mortality rate and EIP are large, their combined effect is large to reduce
the relative vectorial capacity. Thus these three parameters need to be measured as
accurately as possible and to be carefully checked before being used in the modeling.
As we have seen, it is quite challenging to incorporate the temperature effect in dengue
transmission either through the basic or effective reproduction number, or through the
vectorial capacity. It is a balance between building a sophisticated model that accounts for
the detailed vector behavior and using a simple model to give clear insight into the effect of
each important factor. The two sub-sections (2.3 and 2.4) in Methods have shown the two
ways of describing dengue suitability based on temperature.
In the sophisticated model that takes into account of seasonality, the yearly temperature
variation is incorporated as an separate term RE in the effective reproduction number,
Re(t) = R1 + RE.
Here RE describes the effect of seasonality through mosquito Egg’s population (see Eq. (221)), which is zero at low dengue season – winter, and non-zero at high dengue season summer. At t =0, R1 = R0 for a totally susceptible population and Re(t) = R1 = R0 when no egg
hatching, such as in winter. Thus, the effective reproduction number Re will be used instead
of R0 as the criterion for epidemic.
This type of modeling is important since it is relevant to Europe’s temperate climate that egg
overwinter is the most crucial survival mechanism for the mosquito population. The
disadvantage of this model is that no simple insight can be drawn without computer
simulation involved. In addition, it has assumed temperature independent parameters in R1.
As we have seen in the Results section, every vector parameter in V/m depends on
temperature for both its mean value T and its daily fluctuation amplitude, DTR. Thus, even
this sophisticated model cannot take into account all the important factors about vector
behavior. Also it relies on seasonality as to be similar in different places, while the seasonality
of temperature between different climates can vary greatly. For example, some climate for
example may have two transmission seasons per year. A more universal representation of
seasonality may be based on the seasonality in observational temperature time series.
In contrast, the simple SIS model in section 2.2 has illustrated the important quantity in
describing vector-borne infection, vectorial capacity, V. Using this quantity and the
parameters found through literature search, the result of dengue relative vectorial capacity
based on temperature was calculated.
In choosing the type of modeling, SIS was chosen although two types of models are both used
for infectious diseases, SIR and SIR. SIR is used for the situation where only one type of
dengue virus exists and where recovered becomes immune, and SIS is used for the situation
where more than one type of virus coexists and where recovered from one type virus is still
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susceptible to other types of viruses. In principle, each virus type should be considered
separately since all the vector parameters are in principle different for each virus, therefore
only SIR model may be used. However, since the data is limited for different viruses in
dengue vector parameters, the model did not account for the detailed path difference
between different viruses. In fact, in terms of initial threshold criterion for an epidemic to
spread in a totally susceptible population, it does not matter which model is used – both
leads to the same expression of vectorial capacity. However, different models will affect the
susceptible and infected population dynamics, which leads to differences in the length and
sustainability of epidemic. It affects directly the susceptible population and will in turn affect
the effective reproduction number Re. Thus, SIR model used for one type of virus runs out of
susceptible quickly and the epidemic dies out if no large birth or migration of people into the
infected area to increase the susceptible population. On the other hand, if the dengue
infection involves more than one type of viruses at the same time, SIS model allows resupply
of the susceptible and makes possible a sustained epidemic or slow decrease of the
susceptible population and correspondingly a slow decrease of the epidemic.
Limitations of this study and suggestions
The calculation of V/m over DTR – diurnal temperature range, depends on five vector
parameters. At T= 26 0C, the DTR dependent vector data were found for four out of five
parameters: bH , bM, n & µM, and no data on how a depends on DTR. At T= 14 0C, the data
were found for only two out of the five parameters: bH and bM, and no data on the dependence
of a, n and µM on DTR. Thus, V/m dependent on DTR is not perfect. In addition, at lower
mean temperature, the daily biting rate a was extrapolated based on fitting to measured data
on higher temperatures. Thus the results presented here can be improved if there is more
data on parameters available. Therefore, this study suggests that a systematic review of
dengue vector parameters is in great need to support modeling studies.
The daily temperature fluctuation modeling used a thermodynamic model from theoretical
biology. Due to time limit of this thesis, I could not manage to learn this model to incorporate
the DTR dependence of parameters on all of the temperatures. If the DTR dependent values
of all the vector parameters were available, it would be possible to produce a 3-D map for
V/m as a function of T and DTR.
All the results presented here are for mosquito Aedes aegypti. As mentioned earlier, the type
that is threatening Europe is Aedes albopictus whose capacity in transmitting dengue is more
important to be modeled. Since it is not the dominant mosquito in tropical and subtropical
regions for dengue, there are less studies about it and less data available. Even though the
few papers that I have found including Albopictus mosquito data showed that the mortality
rate is very similar between these two types of mosquitoes, the overall effect of all the
parameters on vectorial capacity is not necessarily the same between the two types of
mosquitoes. Studies have shown that Aedes aegypti is more likely to be found in dry and
warm places while Aedes albopictus are more likely to be found in cool and wet places
(Juliano et al., 2002; Alto and Juliano, 2001). Furthermore, in transmission of viruses, Aedes
aegypti mainly transmits through horizontal channel – human hosts, while Aedes albopictus
does appear to transmit the dengue virus through both horizontal and vertical transmission.
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Given the differences between the two dengue vectors, further study is needed to model more
specifically dengue suitability of vector Aedes albopictus.
Humidity and rainfall are important climate factors that have not been considered in all the
mathematical models that I have encountered. The reason is that it is local effect and is hard
to incorporate. Small places holding water such as old tires and waste containers are
important breeding places for dengue vectors. Not like temperature, there is not general data
on the local humidity and the relationship between the parameters and rainfall and humidity.
As mentioned earlier, the vector parameters depend on virus types (Gubler et al., 1979) and
dosage (Watts, 1987) and many geographic strains of dengue viruses exists. They can cause
different reaction in people. For example, the South American dengue virus strain does not
cause bleeding as the Asian virus strain does. It demands more understanding of the
interaction between vector and virus, and complete data input in order to set up better
mathematical models.
Mosquito population depends on temperature as well and on local humidity and rainfall. Due
to lack of data and analytical relation between the mosquito population and weather factors,
it is not considered explicitly in the model.
Regarding to the model itself, the seasonality is built in the temperature variations – both
mean value T and DTR – of vector parameters. It can, in principle, be incorporated as a
temperature dependent vector parameter in all the parameters in the expression of V, as it
has been shown in the results section for two mean temperatures, T. Since it is the adult
female mosquitoes that infect human, the effect of egg hatching is to account for the
mosquito population changes: increase in the summer and decrease in the winter. Thus,
seasonality can also be incorporated in the vector to human population ratio, m (= NM/NH),
as a sinusoidal function of time in the basic reproduction number or in vectorial capacity, as
part of the transmission rate as shown in Sec. 2.3.1. Thus, it is possible to use the simple
expression of V as shown in Eq. (2-12) to calculate dengue vectorial capacity dependence on
temperature without the need for a separate term to take into account of seasonality, at least
in the dengue risk assessment of a fully susceptible population.
This study has focused on part of the physical environment – two climate factors on dengue –
mean temperature and diurnal temperature range. However, study has shown that dengue
burden is not only the result of environmental (climate) factor but also socioeconomic and
demographic factors. The fact that similar climate but across two country borders (Texas,
USA and Mexico) has different dengue incidences indicates that other determinants are
important also and need to be considered in modeling (Reiter et al., 2003). The social and
political environment affect both the human’s cope ability through recovery rate to dengue
and mosquito’s population through breading ground availability and human action on adult
mosquito control. These factors, in principle, can be taken into account in both the recovery
rate, γH, and the mosquito to human population ratio, m, for a specific population in the
calculation of the basic reproduction number but not in the relative vectorial capacity.
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Conclusion
This thesis has reviewed the basic and some sophisticated theoretical frameworks in
infectious disease modeling with focus on dengue mathematical modeling. Temperature
effect on dengue transmission was explored from two different models – seasonal and daily
variation. Dengue risk based on temperature is calculated through the quantity – relative
vectorial capacity (vectorial capacity relative to vector to human population ratio, V/m), as a
function of mean temperature and of diurnal temperature range at two different mean
temperatures: 14 and 26 0C. This may represent two different seasons in one place or two
different places in the same season. The result showed that 28-30 0C is the optimal mean
temperature for dengue transmission. Daily temperature variation is a very important
climate factor in dengue transmission. Increasing the daily temperature fluctuation reduces
greatly the gap between cooler and warmer regions in dengue transmission. As global
warming continues with increased temperature and extreme temperature variation, this
result is important in considering dengue risk assessment – the spread and establishment in
current non-transmission areas.
From the Methods section, we have seen that different theoretical frameworks exist with
different complexity and approximations. It is a challenge to incorporate temperature effects
in dengue studies to give a meaningful prediction of epidemic threshold without losing
insight in the jungle of complex mathematics.
From the Results section, it is clear that knowing and choosing the right vector parameters
are important, but challenging aspect of mathematical modeling. Here empirical studies from
biology and zoology are essential. The generalizability of the model depends on the
parameters used and assumptions made. From sensitivity analysis, the vector mortality rate
or the daily survival probability is the most important vector parameter in affecting the value
of vectorial capacity. Thus, its value needs to be chosen carefully in any modeling and to be
measured accurately in any empirical study.
This work has illustrated that a simple model can give important insight into the disease
spreading. A systematic review of the vector parameters is in great need in modeling to
increase its usefulness and generalizability in understanding, predicting and guiding disease
control.
To set up a good model for infectious disease in the face of climate change is quite a challenge.
As an individual, it requires broad knowledge from mathematics, epidemiology, computer
science, biology, meteorology to sociology. As a field, the challenges of mathematical
modeling are in communicating changing assessments of risk in face of great uncertainty.
Population is more mobile so that diseases spread faster and wider. This requires faster
responses and understanding of potential transmission risk areas (Van Kerkhove et al.,
2010). For dengue study specifically, the challenges lie on 1) better natural history and
transmission model, 2) data on vector transmission and virus interaction, and 3)
maintaining balance between simplicity that gives clear insight and complexity that captures
the effect of climate change.
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